Cosmographic analysis of the equation of state of the universe through Pad\'e approximations

Cosmography is used in cosmological data processing in order to constrain the kinematics of the universe in a model-independent way, providing an objective means to evaluate the agreement of a model with observations. In this paper, we extend the conventional methodology of cosmography employing Taylor expansions of observables by an alternative approach using Pad\'e approximations. Due to the superior convergence properties of Pad\'e expansions, it is possible to improve the fitting analysis to obtain numerical values for the parameters of the cosmographic series. From the results, we can derive the equation of state parameter of the universe and its first derivative and thus acquire information about the thermodynamic state of the universe. We carry out statistical analyses using observations of the distance modulus of type 1a supernovae, provided by the union 2.1 compilation of the supernova cosmology project, employing a Markov chain Monte Carlo approach with an implemented Metropolis algorithm. We compare the results of the original Taylor approach to the newly introduced Pad\'e formalism. The analyses show that experimental data constrain the observable universe well, finding an accelerating universe and a positive jerk parameter. We demonstrate that the Pad\'e convergence radii are greater than standard Taylor convergence radii, and infer a lower limit on the acceleration of the universe solely by requiring the positivity of the Pad\'e expansion. We obtain fairly good agreement with the Planck results, confirming the $\Lambda$CDM model at small redshifts, although we cannot exclude a dark energy density varying in time with negligible speed of sound.